Non-linear vibration of initially stressed plates with initial imperfections

Non-linear partial differential equations of vibration for isotropic plates having initial imperfection are derived. The derivation based on the classical plate theory aims to describe non-linear vibration of imperfect plates in a general state of arbitrary initial stresses. Galerkin method is used to reduce the non-linear partial differential equations to ordinary non-linear differential equations. Runge-Kutta method is used to obtain the non-linear and linear frequencies of vibration. A numerical example is presented to discuss the performances of perfect and imperfect plates. The initial stress is taken to be a combination of pure bending stress plus an extension stress in the plane of the plate. It is found that the existence of initial vibration amplitude, initial stress and geometric imperfect may result in a drastic change on the non-linear vibration behavior. The effects of various parameters on the non-linear free vibrations are discussed.     

弹性地基上正交异性板的振动分析

采用微分求积法对非线性弹性地基上的正交异性板的振动响应进行数值模拟。采用微分求积技术将偏微分方程转化为离散特征值问题。数值计算结果与相应文献的结果较为接近。数值计算结果表明,约束刚度、平板高宽比和地基刚度对正交异性板具有很大影响。     

Nonlinear analysis of imperfect laminated thin plates under transverse and in-plane loads and resting on an elastic foundation by a semi-analytical approach

The large deflection and postbuckling behavior of imperfect composite laminated plates exposed to a combined action of transverse loads and in-plane edge compressions and resting on an elastic foundation are investigated in this paper by a semi-analytical approach. The formulations are based on the classical laminated plate theory (CLPT), and include the plate–foundation interaction effects via a two-parameter model (Pasternak-type) from which Winkler elastic foundation can be recovered as a limiting case. The present approach employs a perturbation technique, one-dimensional differential quadrature approximation and the Galerkin procedure to model the nonlinear performance of the plate with arbitrary combination of simply supported, clamped or elastic rotational edge constraints. Studies concerning its accuracy and convergence characteristics are carried out through some numerical illustrations. Effects of foundation stiffness, plate aspect ratio, total number of plies, fiber orientation, initial geometrical imperfection, the character of boundary conditions, and load patterns on the nonlinear behavior of the plate are studied. Typical numerical results are given in dimensionless graphical forms.     

Nonlinear free vibration of tapered Mindlin plates with edges elastically restrained against rotation using DQM

Large amplitude free vibration analyses of tapered Mindlin rectangular plates with elastically restrained against rotation edges are investigated using different differential quadrature method (DQM). The governing equations are based on the first-order shear deformation plate theory in conjunction with Green's strain and von Karman assumption. The spatial derivatives are discretized using DQM and the harmonic balance method is used to transform the resulting differential equations into frequency domain. A direct iterative method is used to solve the nonlinear eigenvalue system of equations. The convergence of the method is shown and their accuracy is demonstrated by comparing the results with those of the limiting cases, i.e. nonlinear free vibration analysis of plates with classical boundary conditions and also linear free vibration analysis of tapered plates. The effects of the elastic restraint coefficient at the edges and the geometrical parameters on the ratio of the nonlinear natural frequency to linear natural frequency of plates with linearly and bi-linearly varying thickness are studied.     

Dynamic stiffness analysis of laminated composite plates

An analysis is presented for the vibration and stability problem of composite laminated plates by using the dynamic stiffness matrix method. A dynamic stiffness matrix is formed by frequency dependent shape functions which are exact solutions of the governing differential equations. It eliminates spatial discretization error and is capable of predicting several natural modes by means of a small number of degrees of freedom. The natural frequencies and buckling loads of composite laminated plates are calculated numerically. The effects of the boundary conditions, the number of layers, the orthotropicity ratio, the side to thickness ratio, and the aspect ratio are studied. It is also illustrated that connected composite plate structures can be handled without difficulty by the present method.     

Nonlinear vibration of initially stressed functionally graded plates

In this paper, nonlinear partial differential equations for the vibration motion of an initially stressed functionally graded plate (FGP) are derived. The formulations are based on the classical plate theory and derived for the nonlinear vibration motion of an FGP in a general state of arbitrary initial stresses. The material properties of the FGP are assumed to vary continuously from one surface to another, according to a simple power law distribution in terms of the constituent volume fractions. The motion of functionally graded ceramic/metal plate is obtained by employing the Galerkin method and then solved by Runge–Kutta method. The initial stress is taken to be a combination of pure bending stress plus an extensional stress in the plane of the ceramic/metal plate. It is found that the frequency responses of nonlinear vibration are sensitive of the state of initial stress, the amplitude of vibration and the volume fraction of FGP.     

一种改进的Pasternak地基模型及层合地基板的解析解

从三维弹性力学基本方程出发,建立了正交异性材料层合板的状态方程,并提出一种改进的Pasternak弹性地基模型,给出了四边简支层合地基板的解析解。此解满足层合板的基本方程和层间连续条件,适合任意厚跨比,计及了所有材料常数,考虑了基底切向接触应力的影响。算例讨论了地基参数对几种地基模型的影响。计算结果表明,随着地基刚度的增大,剪应力的影响是不可忽略的。     

Periodicity in the response of nonlinear plate, under moving mass

The dynamics of nonlinear thin plates under influence of relatively heavy moving masses is considered. By expansion of the solution as a series of mode functions, the governing equations of motion are reduced to an ordinary differential equation for time development of vibration amplitude, which is Duffing's oscillator with time varying coefficients. Through the application of Banach's fixed-point theorem, the periodic solutions are predicted. The method presented in this paper is general so that the response of plate to moving force systems can also be considered.     

Thermal postbuckling of laminated composite skew plates with temperature-dependent properties

Thermal postbuckling and consequently, the role of temperature dependence of material properties on the thermal postbuckling behavior of laminated composite skew plates have not been addressed in literature. Hence, this problem is investigated here. The plate governing equations are based on the first-order shear deformation theory (FSDT) and the geometrical nonlinearity is modeled using Green’s strain tensor in conjunction with the von Karman assumptions. Since the problem is geometrically and physically nonlinear, the differential quadrature method (DQM) as an accurate, simple and computationally efficient numerical tool is adopted to discretize the governing equations and the related boundary conditions. Then, a direct iterative method is employed to obtain the critical temperature (bifurcation point) and consequently the nonlinear equilibrium path (the postbuckling behavior) of symmetrically laminated skew plates. After validating the formulation and the method of solution, the effects of temperature dependence of the material properties on the postbuckling characteristic of laminated skew plates with different skew angle, boundary conditions, length-to-thickness ratio, number of layers and ply layout are investigated.     

A differential quadrature nonlinear free vibration analysis of laminated composite skew thin plates

Using a differential quadrature (DQ) method, large amplitude free vibration analysis of laminated composite skew thin plates is presented. The governing equations are based on the thin plate theory (TPT) and the geometrical nonlinearity is modeled using Green's strain in conjunction with von Karman assumptions. To cause the impact due to nonlinear terms more significant, in-plane immovable simply supported, clamped and different combinations of them are considered. The effects of different parameters on the convergence and accuracy of the method are studied. The resulted solutions are compared to those from other numerical methods to show the accuracy of the method. Some new results for laminated composite skew plates with different mixed boundary conditions are presented and are compared with those obtained using the first order shear deformation theory based DQ (FSDT-DQ) method. Excellent agreements exist between the solutions of the two approaches but with much lower computational efforts of the present DQ methodology with respect to FSDT-DQ method.     

Bending analysis of thin annular sector plates using extended Kantorovich method

Highly accurate approximate closed-form solution is presented for bending of thin sector plates with clamped edges subjected to uniform and non-uniform loading using the extended Kantorovich method (EKM). Successive application of EKM converts the governing equation which is a forth-order partial differential equation (PDE) to two separate ordinary differential equations (ODE) in terms of r and θ. The obtained ODE systems are then solved iteratively with very fast convergence. In every iteration, exact closed-form solutions are obtained for both ODE systems. It is shown that the method provides sufficiently accurate results not only for deflections but also for stress components. Comparison of the deflection and stresses at various points of the plate shows very good agreement with results of other analytical and numerical analyses.     

Free vibration analysis of variable thickness thin and moderately thick plates with elastically restrained edges by DQM

A differential quadrature (DQ) procedure is developed for free vibration analysis of variable thickness moderately thick plates with edges elastically restrained against translation and rotation. The governing equations are based on the first order shear deformation theory and the rotary inertia effects are considered. Comparisons with known thin plate and uniform thickness Mindlin plate solutions are carried out to verify the applicability and accuracy of the analysis. Plates with linear or nonlinear varying thickness in one or two directions can be considered. It is demonstrated that using this DQ procedure, classical boundary conditions such as simply supported, clamped and free edges for variable thickness, thin as well as moderately thick plates can be simulated without any numerical difficulties. The effects of different combinations of constraints at edges, aspect ratio, thickness-to-length ratio, and stiffness parameter values on accuracy and convergence behaviors of the plates are presented. Some new results are presented for bi-linearly variable thickness plates with elastically restrained edges.     

轴压组合叠合梁自由振动的动态刚度分析

采用动态刚度分析方法,对自由振动和轴压组合叠合梁的屈曲性能进行研究。在公式中考虑了轴力、泊松效应、轴向变形、剪切变形和转动惯量的影响。通过直接求解轴压组合叠合梁自由振动的控制微分方程,建立严格的动态刚度矩阵,并用该矩阵对已有文献中其他组合梁进行计算比较,证明了动态刚度矩阵的适应性。同时,论证了轴力和边界条件对自然频率、屈曲荷载和叠合梁模式的影响。     

Eccentric low-velocity impact analysis of transversely graded plates with Winkler-type elastic foundations and fully or partially supported edges

Impact analysis of plates with either elastic foundations or partially supported edges has not been accomplished, even for the homogeneous isotropic plates. In the present paper, eccentric low-velocity impact analysis of a functionally graded plate with fully/partially supported edges and a Winkler-type elastic foundation is performed. In contrast to the available researches, stiffness of the underneath layers is also taken into account in determination of the apparent stiffness of the contact region, employing Mori–Tanaka micromechanics-based model instead of the traditional rule of mixtures. The governing equations are derived based on the principle of minimum total potential energy and the first-order shear-deformation plate theory. Some of the included novelties are: modification of the contact law, incorporation of effects of the elastic foundation, considering the material heterogeneity and partially supporting the edges, and proposing various semi-analytical and (Galerkin-type) numerical solutions. Numerical time integration schemes are used to trace time variations of the responses. A sensitivity analysis is performed to investigate influences of the specifications of the plate and the indenter, the eccentricity, and the supporting length ratio on time histories of the indentation and contact force.     

弹性地基上厚板的计算分析

以弹性地基上Reissner板为研究对象,地基采用双参数模型,把地基效应归并到厚板的弯曲微分方程中。利用导出的双参数地基上厚板弯曲问题的基本解,从虚功原理出发,依据胡海昌的中厚板弯曲理论,推导出三个广义位移表示的边界积分方程。适用于任意边界条件、任意形状及任意荷载的薄板及中厚板的弯曲问题。文中给出了算例,并对各种参数变化进行了分析,获得了满意的结果。     

弹性地基上功能分级梁的二维弹性解

基于二维弹性理论,提出了Winkler-Paster-nak弹性地基上功能分级梁弯曲和自由振动的精确解。假设梁是正交各向异性,沿厚度方向材料性能指数变化。通过扩展状态变量为无穷三角级数,调节偏微分方程系统,最终采用状态空间法使这一问题得到解决。对几个参数的影响,如梯度指数、纵横比和功能梯度材料梁基本参数的力学性能进行研究。所提出的数值结果,为这类梁的分析提供了依据。     

Approximate elliptical integral solution for the large amplitude free vibration of a rectangular single mode plate backed by a multi-acoustic mode cavity

The nonlinear structural–acoustic problem considered in this study is the large amplitude free vibration of a rectangular elastic plate backed by a cavity. Very few classical solutions for this nonlinear structural–acoustic problem have been developed, although there are many for nonlinear plate or linear structural–acoustic problems. Thus, the main contributions of this study paper include (1) a concise multi-acoustic single structural modal formulation that is derived from two coupled partial differential equations representing the nonlinear structural free vibration and the acoustic pressure induced and (2) the approximate elliptical integral solution that is obtained by solving one residual equation only, and well agrees with that obtained from a harmonic balance finite element analysis. It is found that the natural frequency convergences with the increase in the numbers of acoustic modes and harmonic terms, and the effects of vibration amplitude, air cavity depth, and aspect ratio on the nonlinear natural frequency are also examined.     

Vibration of cross-ply laminated composite plates subjected to initial in-plane stresses

Natural frequencies, modal displacements and stresses of cross-ply laminated composite plates subjected to initial in-plane stresses are analyzed by taking into account the effects of higher-order deformations and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for rectangular laminates is derived through Hamilton’s principle. Several sets of truncated approximate theories can be derived to solve the eigenvalue problems of a simply supported laminated plate. After examining the convergence properties of the lowest natural frequency, only the numerical results for M=5, which are considered to be sufficient with respect to the accuracy of solutions, are presented. Numerical results are compared with those of the published existing three-dimensional theory and FEM solutions. The modal displacement and stress distributions in the thickness direction are plotted in figures. The buckling stresses can be obtained in terms of the natural frequencies of the laminates without initial in-plane stresses.     

移动荷载下正交各向异性地基无限大板的动力响应

以薄板理论和弹性动力学理论为前提,以位移分量为基本未知量,建立了直角坐标系下的移动谐振荷载作用下正交各向异性地基上覆无限大弹性板的力学模型和动力微分方程;然后用坐标变换和Fourier积分变换,且引入边界条件,推导了移动荷载作用下无限大板的挠度和薄板与地基之间的接触应力的积分形式解。基于推导的理论方法,编制了相应的计算程序,并对薄板表面作用线性谐振荷载问题进行了算例分析,验证了方法的正确性。最后,对移动谐振荷载作用下公路路面板的动力响应进行了参数分析,研究了土体参数、板参数、荷载速度、荷载频率对其影响规律。结果表明:土体的各向异性、板厚、板的弹性模量、荷载移动的速度和振动频率对板动力响应影响很大。     

Elastic stability of orthotropic plates

A semi-numerical method is presented for the analysis of orthotropic rectangular plates subjected to uniform, linearly-varying and partial inplane loads. The solution procedure is based on the classical method of separation of variables. The basic functions are chosen as the eigenfunctions for straight prismatic beams in free vibration. The two-dimensional governing partial differential equation is first reduced to an ordinary differential equation. The classical unidirectional finite difference method is then employed to solve the resulting ordinary differential equation. Results are presented for plates with different edge and loading conditions.